Sixteen Eisenstein series
type: Item
instance of: scholarly article
title: Sixteen Eisenstein series
zbMATH Open document ID: 1195.11048
author: Shaun Cooper
author: Heung Yeung Lam
published in: The Ramanujan Journal
publication date: 2009-11-23T00:00:00Z
review text: Using the theory of elliptic functions the authors prove several identities which have as particular cases earlier formulas of Ramanujan. For example: \[ E_{2n}-4\sum_{j=1}^\infty\frac{(-1)^j(2j-1)^{2n}q^{2j-1}}{1-q^{2j-1}}=z^{2n+1}p_n(x),\quad n=0,1,2,3,\ldots, \] where \(E_n\) are the Euler numbers defined by \[ \frac{2e^u}{e^{2u}+1}=\frac{E_nu^n}{n!}, \] \(q=\exp(i\pi\tau)\), \(\text{Im}(\tau)>0\), \(z=z(q)=\left(\sum_{n=-\infty}^\infty q^{n^2}\right)^2\), and \(p_n(x)\) is a polynomial of degree \(n\) in \(x\) with rational coefficients.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 5636934
MaRDI profile type: Publication
full work available at URL: s11139-007-9039-3
OpenAlex ID: W2074014605
cites work: Q4000402
cites work: Q5485643
cites work: On Eisenstein series and ∑_{𝑚,𝑛=-∞}^{∞}𝑞^{𝑚²+𝑚𝑛+2𝑛²}
cites work: Q3695591
cites work: On sums of an even number of squares, and an even number of triangular numbers: An elementary approach based on Ramanujan's \(_1\psi_1\) summation formula
cites work: Cubic theta functions.
cites work: On Lerch's Transcendant and the Basic Bilateral Hypergeometric Series 2 ψ2
cites work: Elliptic functions and applications
cites work: On Summation of Series of Hyperbolic Functions. II
cites work: Generalization of Certain Summations Due to Ramanujan
cites work: Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions
cites work: Q5513003
cites work: The numerical value of certain series
cites work: Q4237087
cites work: The Summation of Series of Hyperbolic Functions
cites work: Some Infinite Series of Exponential and Hyperbolic Functions
DOI: 10.1007/S11139-007-9039-3
description: scientific article; zbMATH DE number 5636934